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A295357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 11
1, 3, 5, 20, 42, 83, 149, 259, 438, 730, 1204, 1973, 3219, 5237, 8504, 13792, 22350, 36200, 58612, 94878, 153559, 248509, 402143, 650730, 1052954, 1703768, 2756809, 4460667, 7217569, 11678332, 18896000, 30574434, 49470539 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Guide to related sequences:

***** Part 1:  initial values are a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6

A295357: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3)

A295358: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3)

A295359: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2*b(n-3)

A295360: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 3*b(n-3)

A295361: a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) - 3*b(n-3)

A295362: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - b(n-3)

***** Part 2: initial values as shown

A295363: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4

A295364: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4

A295365: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6

A295366: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6

A295367: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4

For all of these sequences, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

LINKS

Table of n, a(n) for n=0..32.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that

b(3) = 7 (least "new number")

a(3) = a(1) + a(0) + b(2) + b(1) + b(0) = 20

Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...)

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; b[2] = 6;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + b[n - 3];

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

z = 32; u = Table[a[n], {n, 0, z}]   (* A295357 *)

v = Table[b[n], {n, 0, 10}]  (* complement *)

CROSSREFS

Cf. A001622, A293076, A294532.

Sequence in context: A231627 A261116 A295361 * A076149 A133102 A197156

Adjacent sequences:  A295354 A295355 A295356 * A295358 A295359 A295360

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 21 2017

STATUS

approved

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Last modified March 31 23:42 EDT 2020. Contains 333152 sequences. (Running on oeis4.)